Jtam-A4.dvi


JOURNAL OF THEORETICAL

AND APPLIED MECHANICS

55, 1, pp. 129-139, Warsaw 2017
DOI: 10.15632/jtam-pl.55.1.129

AN ENERGY-BASED METHOD IN PHENOMENOLOGICAL DESCRIPTION

OF MECHANICAL PROPERTIES OF NONLINEAR MATERIALS UNDER

PLANE STRESS

Tadeusz Wegner, Dariusz Kurpisz

Poznan University of Technology, Institute of Applied Mechanics, Poznań, Poland

e-mail: dariusz.kurpisz@put.poznan.pl

A method based on energy is a very useful tool for description of mechanical properties of
materials. In the current paper, on the base of geometrical interpretation of a deformation
process, the strain energy density function for isotropic nonlinear materials has been con-
structed.On account of hydrostatic interpretation of the volumetric deformation, the elastic
part of energyhas been extracted.The initiation of the damageprocess due to plastic flowof
the material under plane stress has been determined and the stability conditions have been
formulated byusing in the stability analysis the strain energydensity function in addition to
Sylvester’s theoremandassumption of zero volume changeduringpure plastic deformations.
This concept is an original part of thework and continuationof the investigations previously
carried out byWegner andKurpisz. The theoretical investigations have been illustrated on
the example of aluminium.

Keywords: energy-based method, nonlinear material, phenomenological description, strain
energy density function, Sylvester’s theorem

1. Introduction

The phenomenological description of mechanical properties of nonlinear materials is interesting
due to possibility of its application to real engineering structures. The knowledge aboutmecha-
nisms of deformation under the influence of external loads and the relations between them can
be very important in the design process. There is a lot of publications devoted to themodelling
of mechanical properties based both on the ground of microstructure relations in the material
(multi-scale modellingmethods), see for example Silva et al. (2007), Terada et al. (2008), Speirs
et al. (2008), or on a phenomenological concept, see for exampleWegner (2000, 2005). Because
we consider mechanical properties on the base of a real experiment in which the object (mate-
rial sample) andmeasured properties are usually inmacro-scale, so using the phenomenological
method as the way of modelling of the process, which is directly connected with the experi-
ment, is recommended. From this point of view, very interesting are methods based on energy,
which were used by Petryk (1985, 1991), Schroder and Neff (2003), Wegner (2000, 2005, 2009)
and Dargazany et al. (2012). Here, the necessary tool of description of mechanical properties
of materials is the strain energy density function. It can be introduced in many different ways,
for example as a direct function of invariants of the deformation state, seeWegner (1999, 2009)
and Schroder and Neff (2003) or on the base of geometrical interpretation of the deformation
process, seeWegner andKurpisz (2009).Wegner andKurpisz (2013), using a phenomenological
approach and basing on the strain energy density function, investigated the damage process of
a metal foam.

Material damage as a dissipativemechanismdescribed in formof free energy per unit volume
considered as a thermodynamic potential was taken into account byCimetiere et al. (2005). The



130 T.Wegner, D. Kurpisz

authors, in amajor way, split the energy into two parts. The first is an elastic (reversible) part,
whereas the second (dissipative) includes, among others, the hardening effect. Because each of
these two parts depends on internal variables, so the internal variables influence the damage
threshold.

Gajewska and Maciejewska (2005) investigated the influence of internal restrictions (con-
nected with the type of the material, for example isotropic or anisotropic one) in form of limit
conditions based on energy of anisotropicmaterials. Such conditions can be interpreted as diffe-
rent yield conditions. Itwas shown that as longas the energy scalar productwasdefinedproperly
in the elastic range, the limit condition having the energy-based interpretation could be found.
A much more interesting case takes place when we have nonlinear-elasticity, what implies the
necessity of modification of the limit condition.

In the further part of the current paper, on the base of geometrical interpretation, the strain
energy density function of a nonlinear material will be introduced and used for formulation of
stability conditions due to the damage process, what is an alternative point of view to that
presented by Gajewska andMaciejewska (2005).

2. Geometrical interpretation of the deformation state – the basic equation

To introduce a geometrical interpretation of the deformation process, let us take into account
the following assumptions:

• The material is isotropic and nonlinear, so the mechanical properties are the same in all
directions, but the relations between stress and strain can not be described in form of
classic Hook’s law (for linear materials).

• The loading process is static, which means that dynamic effects can be neglected.
• The dissipated part of energy includes thermal and plastic deformation energy.
• Thematerial is under plane stress.
• The longitudinal deformation coefficient is a function of the deformation state due to
nonlinear properties of the material.

• The principal stress and strain directions are the same due to material isotropy.

Every deformation process can be interpreted as a deformation path C, which is located in
space of principal deformation state components. Every point of such path is one deformation
state. So a change of deformation due to a change of external loads (change of principal stress
components) implies a displacement along the path

C : εi = ε
k
i t for i=1,2,3 (2.1)

where εki are the final deformation state components, and t∈T = 〈0,1〉 is a parameter.
The relation between stress and strain in every point of the deformation path takes the form

of generalized Hook’s law

ε1 =
σ1(t)

Ẽ(ε1)
− ṽ(ε2)

σ2(t)

Ẽ(ε2)
− ṽ(ε3)

σ3(t)

Ẽ(ε3)

ε2 =
σ2(t)

Ẽ(ε2)
− ṽ(ε1)

σ1(t)

Ẽ(ε1)
− ṽ(ε3)

σ3(t)

Ẽ(ε3)

ε3 =
σ3(t)

Ẽ(ε3)
− ṽ(ε1)

σ1(t)

Ẽ(ε1)
− ṽ(ε2)

σ2(t)

Ẽ(ε2)

(2.2)



An energy-based method in phenomenological description... 131

Fig. 1. Deformation paths for a triaxial and plane stress state in a material

where longitudinal and transversal deformation coefficients follow from experimental characte-
ristics σ(ε), εt(ε) (εt is transversal deformation) obtained from a uniaxial tension test and take
respectively the form

Ẽ(ε)=
σ

ε
ṽ(ε)=−

εt
ε

(2.3)

Because both stress and strain are changeable (along path Cf) in time t ∈ 〈0,1〉 of the
deformation process, then the density of deformationwork from the initial state εi =0 for t=0
to the final state εi = ε

k
i for t=1 can be expressed as

WC(εk1,ε
k
2,ε
k
3)=

∫

Cf

3∑

i=1

σi dεi =

1∫

0

3∑

i=1

σi(t)ε
′

i(t) dt (2.4)

where on the basis of (2.2) for i=1,2,3

σi(t)= Ẽ(εi)

εi
3∏
l=1
[1+ ṽ(εl)]+ [1+ ṽ(εi)]

3∑
l=1
ṽ(εl)(εl −εi)+ 1+ṽ(εi)ṽ(εi)

3∏
l=1
ṽ(εl)

(
3∑
l=1
εl−3εi

)

3∏
l=1
[1+ ṽ(εl)]− [1+ ṽ(εi)]2

(
3∑
l=1
ṽ(εl)− ṽ(εi)+ 2ṽ(εi)

3∏
l=1
ṽ(εl)

)

(2.5)

In particular, if we assume the plane stress in plane 1O2, then the stress components in the
third direction are equal zero, and relations (2.4) and (2.5) simplify respectively to

WC(εk1,ε
k
2)=

∫

Cf

2∑

i=1

σi dεi =

1∫

0

2∑

i=1

σi(t)ε
′

i(t) dt (2.6)

and

σ1(t)= Ẽ(ε1)
ε1+ε2ṽ(ε2)

1− ṽ(ε1)ṽ(ε2)
σ2(t)= Ẽ(ε2)

ε2+ε1ṽ(ε1)

1− ṽ(ε1)ṽ(ε2)
(2.7)



132 T.Wegner, D. Kurpisz

The strain component in the passive (third) direction takes the form

ε3 =−ṽ(ε1)
ε1+ε2ṽ(ε2)

1− ṽ(ε1)ṽ(ε2)
− ṽ(ε2)

ε2+ε1ṽ(ε1)

1− ṽ(ε1)ṽ(ε2)
(2.8)

In the geometrical interpretation, function (2.6) specifies the values of deformation work
density (2.4), which are defined in the space of three-dimensional deformation to present a
strain energy density distribution function plot (Fig. 5) along section surface (2.8).

3. Extraction of the volumetric part of energy

Let us consider purely volumetric deformations. Such type of deformations takes place if a
material is under influence of hydrostatic pressure. The relation between deformation state
components and stress state components according to hydrostatic pressure takes the form

εV1 =
ks

Ẽ(εV1 )
− ṽ(εV2 )

ks

Ẽ(εV2 )
− ṽ(εV3 )

ks

Ẽ(εV3 )

εV2 =
ks

Ẽ(εV2 )
− ṽ(εV1 )

ks

Ẽ(εV1 )
− ṽ(εV3 )

ks

Ẽ(εV3 )

εV3 =
ks

Ẽ(εV3 )
− ṽ(εV1 )

ks

Ẽ(εV1 )
− ṽ(εV2 )

ks

Ẽ(εV2 )

(3.1)

where s∈T and hence, due to symmetry

εV1 = ε
V
2 = ε

V
3 = ε

V (3.2)

where εV satisfies the equation

εV =
1−2ṽ(εV )
Ẽ(εV )

ks=β(s) (3.3)

and s is a non-dimensional parameter: the ratio of hydrostatic pressure to the basic value k,
ks is the current value of hydrostatic pressure.
The above relation does not provide the information about the connection between the

current value of deformation (point of path C) and its volumetric part (point of path CV ), see
the picture below.

Fig. 2. Relation between the deformation pathC and the path due to pure volumetric deformationCV

To determinate this relation, we have to take into account two analytical descriptions of a
volume change in an elementary piece of the material.
The first way

∆V

V0
=
3∏

i=1

[1+εi(t)]−1 (3.4)



An energy-based method in phenomenological description... 133

where ε1(t), ε2(t) and ε3(t) are deformation components of the pathC.
The second way explores hydrostatic pressure

Θ=
∆V

V0
=
3∏

i=1

[1+β(s)]−1= [1+β(s)]3−1 (3.5)

whereΘ is the relative volume change.
After comparison of the right-hand sides of equations (3.4) and (3.5), we receive

s=h(t)=β−1
(
3

√√√√
3∏

i=1

[1+εi(t)]−1
)

(3.6)

and after substitution into (3.3)

εV = 3

√√√√
3∏

i=1

[1+εi(t)]−1 (3.7)

Hence, we can write that

WC
V

=

∫

CV

σ1 dε
V
1 +

∫

CV

σ2 dε
V
2 +

∫

CV

σ3 dε
V
3 =

3∑

i=1

∫

CV

σi dε
V (3.8)

where σi (i=1,2,3) are solutions to the system of equations (2.2) given in form (2.5).

4. Stability conditions for the material under plane stress

Thematerial is in the stable state of equilibrium if every change of the deformation state needs
work to be done by external loads. So, in other words, we say aboutmaterial stability when the
strain energy density function is convex. In an analytical form, it can be written as

δ2WC =
3∑

i=1

3∑

j=1

∂2WC

∂εki∂ε
k
j

δεkiδε
k
j > 0 (4.1)

where δ2 denotes the second order variation of the functionWC.
In the case of plane stress, the deformationworkWC can be interpreted as a function of two

deformation state components, however from the other side, the strain energy density depends
on three deformation state components and the sign of its second order variation of the strain
energy is strictly connectedwith the threevariation increments of deformation state components.
Hence, on the base of Sylvester’s theorem, we receive

∣∣∣∣∣∣∣∣∣∣∣∣∣∣

∂2WC

∂(εk1)
2

∂2WC

∂εk1∂ε
k
2

∂2WC

∂εk1∂ε
k
3

∂2WC

∂εk1∂ε
k
2

∂2WC

∂(εk2)
2

∂2WC

∂εk2∂ε
k
3

∂2WC

∂εk1∂ε
k
3

∂2WC

∂εk2∂ε
k
3

∂2WC

∂(εk3)
2

∣∣∣∣∣∣∣∣∣∣∣∣∣∣

> 0

∣∣∣∣∣∣∣∣∣

∂2WC

∂(εk1)
2

∂2WC

∂εk1∂ε
k
2

∂2WC

∂εk1∂ε
k
2

∂2WC

∂(εk2)
2

∣∣∣∣∣∣∣∣∣

> 0

∣∣∣∣∣∣∣∣∣

∂2WC

∂(εk1)
2

∂2WC

∂εk1∂ε
k
3

∂2WC

∂εk1∂ε
k
3

∂2WC

∂(εk3)
2

∣∣∣∣∣∣∣∣∣

> 0

∣∣∣∣∣∣∣∣∣

∂2WC

∂(εk2)
2

∂2WC

∂εk2∂ε
k
3

∂2WC

∂εk2∂ε
k
3

∂2WC

∂(εk3)
2

∣∣∣∣∣∣∣∣∣

> 0

∣∣∣∣∣
∂2WC

∂(εk1)
2

∣∣∣∣∣ > 0
∣∣∣∣∣
∂2WC

∂(εk2)
2

∣∣∣∣∣ > 0
∣∣∣∣∣
∂2WC

∂(εk3)
2

∣∣∣∣∣ > 0

(4.2)



134 T.Wegner, D. Kurpisz

Inequality (4.1) implicates a system of six nonlinear inequalities (4.2) which allow us to draft
the region of material stability. Because the plastic deformation leads to permanent loss of the
element shape, so very important is the knowledge about material deformation due to plastic
flow.This type of phenomenon takes place if during the deformation process, the volume change
of an elementary piece of the material is unchanging.
On the basis of (A.15, see Appendix), the second order variation takes the form

δ2WV=const =
3∑

i=1

3∑

j=1

(
∂2WC

∂εki∂ε
k
j

− 1
3

3∑

k=1

σk
3
√
A1A2A3

∂Ai

∂εkj

)
δεkiδε

k
j

=
3∑

i=1

3∑

j=1

Bijδε
k
iδε
k
j

(4.3)

whereA1,A2,A3 are given as in Appendix (A.3), and σk are expressed by using (2.5).

Hence, stability assumption (4.3) takes respectively the form

∣∣∣∣∣∣∣

B11 B12 B13
B21 B22 B23
B31 B32 B33

∣∣∣∣∣∣∣
> 0

∣∣∣∣∣
B11 B12
B21 B22

∣∣∣∣∣ > 0

∣∣∣∣∣
B11 B13
B31 B33

∣∣∣∣∣ > 0
∣∣∣∣∣
B22 B23
B32 B33

∣∣∣∣∣ > 0

|B11|> 0 |B22|> 0 |B33|> 0

(4.4)

In the case of a plane state of stress, so when σ3 = 0, ε
k
3 must to be replaced by relation

(2.8).

5. Example

As an example of using theoretical investigations, aluminum in plain stress has been taken. The
experimental plots of material characteristics are presented in Figs. 3 and 4.

Fig. 3. Experimental relation between stress and strain (in a uniaxial tensile test) and its approximation
in the nonlinear range (red line)

However, the precision of approximation of the experimental characteristic between stress
and strain is not sufficient for very small (closed to zero) deformations, it is very accurate in the
range ε∈ 〈0.0023;0.0033〉, where there exists danger of the appearance of plastic flow. Because
in the analytical assessment of the limit surface the second order partial derivatives of the strain



An energy-based method in phenomenological description... 135

Fig. 4. Experimental plot of transversal deformation coefficient (relation between the ratio of
transversal to longitudinal deformation and longitudinal deformation) and its approximation (red line)

energy density function are very important, then much more important is the accuracy of the
approximation for ε∈ 〈0.0023;0.0033〉 than for ε∈ 〈0;0.0023〉.
Analytical approximations of the above characteristics (σ [MPa], ṽ(ε) [MPa], Ẽ(ε) [MPa])

can be written respectively as

σ=−11100000ε2 +86700ε+9 for ε∈ 〈0;0.0033〉

ṽ(ε)=





0.317 for ε∈ 〈0;0.0032〉
0.366

π
arctan

(
2601ε
ε−0.0032
0.02−ε

)
+0.317 for ε> 0.0032

(5.1)

Hence from (2.3)1 and (5.1), we have

Ẽ(ε)=−11100000ε+86700+ 9
ε

for ε∈ 〈0;0.0033〉 (5.2)

The strain energy density function (WC [MJ/m3]) due to (2.5), (5.1) and (5.2) takes the
form

WC(εk1,ε
k
2,ε
k
3)=

1∫

0

3∑

i=1

σi(t)ε
′

i(t) dt=

1∫

0

3∑

i=1

σi(t)ε
k
i dt

=
a1(1+ c1)−3a1c1
3(1+ c1)(1−2c1)

3∑

i=1

(εki )
3+
b1(1+ c1)−3b1c1
2(1+ c1)(1−2c1)

3∑

i=1

(εki )
2

+
a1c1

3(1+c1)(1−2c1)
3∑

i=1

(εki )
2
3∑

l=1

εkl +
b1c1

2(1+ c1)(1−2c1)

(
3∑

i=1

εki

)2
+

1

1−2c1

3∑

i=1

εki

(5.3)

where a1 =−11100000, b1 =86700, c1 =0.317.
If we would like to represent and plot in an easy way the strain energy density func-

tion for a plain state of stress, then we have to substitute (on the basis of relation (2.8))
εk3 =−[c/(1− c)](εk1 +εk2) into relation (5.4). After transformations, we receive

WC(εk1,ε
k
2)=−36815920[(εk1)3+(εk2)3]+431343.3[(εk1)2+(εk2)2]

+117.94(εk1 +ε
k
2)−331343284[(εk1)2εk2 +εk1(εk2)2]+7764179.1εk1εk2

(5.4)

for εk1 ∈ 〈0;0.0033〉, εk2 ∈ 〈0;0.0033〉.
If we apply relation (5.3) to stability assumptions (4.5) then we receive stability regions

shown in Fig. 6.



136 T.Wegner, D. Kurpisz

Fig. 5. The strain energy density function

Fig. 6. Region of stability according to assumptions (4.5) in space of the deformation (a) and the
stress (b) components

The above plots are obtained from the major form of stability assumptions (formulated for
a three-axial state of stress), after substituting the relation εk3 =−[c(1− c)](εk1 + εk2), which is
true in the case of plain stress.

6. Conclusions

• The strain energy density function is a sufficient tool in the description of mechanicals
properties of nonlinear material and necessary for stability analysis.

• Extraction of the volumetric part of energy is possible due to hydrostatic interpretation.
• The constant volume assumption plays an important role in formulation of stability con-
ditions due to plastic flow.

• The limit surfaces are convex (see. Figs. 6) and are comparable with the known limit
surfaces for linear-elastic materials.

A. Appendix

The material is in the stable state of equilibrium if every change of current deformation state
needs awork by external loads. So, if we take into account a very small fluctuation of the current
deformation state in three principal directions then relation (4.1) has to be satisfied.



An energy-based method in phenomenological description... 137

Theplastic flowtakesplace in thecasewhenan increment involumeequals zero, see (Wegner,
2000).
If l1, l2 and l3 are dimensions of an elementary volume piece of the material in an unloaded

state then the ratio of volume change to its virgin state can be written as

θ=
∆V

V0
=

3∏
i=1
(li+∆li)−

3∏
i=1
li

3∏
i=1
li

=
3∏

i=1

(1+εi)−1 (A.1)

and hence

δθ= δ

[
3∏

i=1

(1+εi)

]
=
3∑

i=1

Aiδεi (A.2)

where

A1 =(1+ε2)(1+ε3) A2 =(1+ε1)(1+ε3) A3 =(1+ε1)(1+ε2) (A.3)

Ifwe take into considerationspurelyvolumetricdeformations then εi = ε
V ,whichby substitution

into (A.1) implies

θ=(1+εV )3−1 ⇒ δθ=3(1+εV )2δεV ⇒ δεV =
δθ

3(1+εV )2
(A.4)

The first order variation of the strain energy density function according to plastic flow can be
divided as follows

δW(V=const) =σ1δε
(V=const)
1 +σ2δε

(V=const)
2 +σ3δε

(V=const)
3

δε
(V=const)
i = δεi− δε

V
(A.5)

In the case of the limit surface according to plastic flow, we do not observe a change in the
volume, so

δWV=const = δWC − δWCV = δWs (A.6)

where on the basis of (3.8), we have

δWC
V

=
3∑

i=1

σiδε
V (A.7)

Hence, taking into account (A.4)

δ2WC
V

=
1

3

3∑

i=1

σi
[ δ2θ
(1+εV )2

− 2δθδε
V

(1+εV )3

]
(A.8)

In the absence of volumetric deformations

δθ= δεV =0 (A.9)

we have

δ2WC
V

=
1

3

3∑

i=1

σi
δ2θ

(1+εV )2
(A.10)



138 T.Wegner, D. Kurpisz

and after using (A.6)

δ2WV=const = δ2WC − δ2WCV =
3∑

i=1

3∑

j=1

∂2WC

∂εi∂εj
δεiδεj −

1

3

3∑

i=1

σi
δ2θ

(1+εV )2
(A.11)

which on the basis of (A.2) gives

δ2WV=const =
3∑

i=1

3∑

j=1

(
∂2WC

∂εi∂εj
−
1

3

3∑

k=1

σk
(1+εV )2

∂Ai
∂εj

)
δεiδεj (A.12)

On the grounds of (A.4)

1+θ=(1+εV )3 ⇒ 1+εV = 3
√
1+θ ⇒ 1+εV = 3

√√√√
3∏

i=1

(1+εi)

⇒ (1+εV )2 = 3
√
A1A2A3

(A.13)

The above relation enables us to write (A.12 in an equivalent form

δ2WV=const =
3∑

i=1

3∑

j=1

(
∂2WC

∂εi∂εj
− 1
3

3∑

k=1

σk
3
√
A1A2A3

∂Ai
∂εj

)
δεiδεj (A.14)

Acknowledgments

This study was supported by DS: 02/21/DSPB/3452.

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Manuscript received November 21, 2014; accepted for print June 15, 2016